Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials

Author(s): Wolfram Koepf; Mohammad Masjed-Jamei
Journal: Proc. Amer. Math. Soc. 135 (2007), 3599-3606.
MSC (2000): Primary 33C45
Posted: June 29, 2007
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. The best-known example of this type is the Hermite functions, i.e., the Hermite polynomials multiplied by $ \exp(-x^2/2)$, which are eigenfunctions of the Fourier transform. In this paper, we introduce two new examples of finite systems of this type and obtain their orthogonality relations. We also estimate a complicated integral and propose a conjecture for a further example of finite orthogonal sequences.

References:

[AAR]
G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)

[Askey1]
R. Askey, Continuous Hahn polynomials, J. Physics A 18, 1985, L1017-L1019. MR 0812420 (87d:33021)

[Askey2]
R. Askey, An integral of Ramanujan and orthogonal polynomials, J. Indian Math. Soc. 51, 1987, 27-36. MR 0988306 (90d:33004)

[AS]
N. M. Atakishiyev and S. K. Suslov, The Hahn and Meixner polynomials of an imaginary argument and some of their applications, J. Physics A 18, 1985, 1583-1596. MR 0796065 (87i:33021)

[Bail]
W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts 32, Cambridge University PFTV, 1935. Reprinted by Hafner Publishing Company, 1972. MR 0185155 (32:2625)

[EMOT]
A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. 2, McGraw-Hill, 1954. MR 0065685 (16:468c)

[Koel]
H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc. 124, 1996, 887-898. MR 1307541 (96f:33018)

[Koep]
W. Koepf, Hypergeometric Summation, Braunschweig/Wiesbaden, Vieweg, 1998. MR 1644447 (2000c:33002)

[Koor1]
T. H. Koornwinder, Special orthogonal polynomial systems mapped onto each other by the Fourier-Jacobi transform, Polynômes Orthogonaux et Applications (C. Brezinski, A. Draux, A. P. Magnus, P. Maroni and A. Ronveaux, Eds.), Lecture Notes Math. 1171, Springer, 1985, 174-183. MR 0838982 (87g:33007)

[Koor2]
T. H. Koornwinder, Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials, Orthogonal Polynomials and their Applications (M. Alfaro, J. S. Dehesa, F. J. Marcellan, J. L. Rubio de Francia and J. Vinuesa, Eds.), Lecture Notes Math. 1329, Springer, 1988, 46-72. MR 0973421 (90b:33024)

[Les]
P. Lesky, Eine Charakterisierung der klassischen kontinuierlichen, diskreten und $ q$-Orthogonalpolynome, Shaker, Aachen, 2005.

[Mas1]
M. Masjed-Jamei, Classical orthogonal polynomials with weight function $ ((ax+b)^2+(cx+d)^2)^{-p}\exp(q\operatorname{arctan} ((ax+b)/(cx+d)))$; $ -\infty<x<\infty$ and a generalization of $ T$ and $ F$ distributions, J. Integral Transforms and Special Functions 15 (2), 2004, 137-153. MR 2053407 (2005b:33011)

[Mas2]
M. Masjed-Jamei, Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation, J. Integral Transforms and Special Functions 13 (2), 2002, 169-190. MR 1915513 (2003i:33011)

[PFTV]
W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Beta Function, $ T$ Student Distribution and $ F$-Distribution, Section 6.2 in Numerical Recipes in Fortran: The Art of Scientific Computing, second edition, Cambridge University Press, Cambridge, 1992, 219-223. MR 1196230 (93i:65001a)

[Rama]
S. Ramanujan, A class of definite integrals, Quarterly J. Math. 48 (1920), 294-310.

[WF]
R. E. Walpole and J. E. Freund, Mathematical Statistics, Prentice-Hall, 1980. MR 0591029 (81k:62002)

[WW]
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1962. MR 0178117 (31:2375)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33C45

Retrieve articles in all Journals with MSC (2000): 33C45

Additional Information:

Wolfram Koepf
Affiliation: Department of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany
Email: koepf@mathematik.uni-kassel.de

Mohammad Masjed-Jamei
Affiliation: Department of Mathematics, K. N. Toosi University of Technology, Sayed Khandan, Jolfa Av., Tehran, Iran
Email: mmjamei@yahoo.com

DOI: 10.1090/S0002-9939-07-08889-2
PII: S 0002-9939(07)08889-2
Keywords: Classical orthogonal polynomials, Fourier transform, hypergeometric functions, Gosper identity, Ramanujan integral
Received by editor(s): January 1, 2006
Received by editor(s) in revised form: August 16, 2006
Posted: June 29, 2007
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2007, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google